Witt groups of abelian categories and perverse sheaves

Abstract

We study the Witt groups $W_{\pm }\left(PervX\right)$ of perverse sheaves on a finite-dimensional topologically stratified space $X$ with even-dimensional strata. We show that $W_{\pm }\left(PervX\right)$ has a canonical decomposition as a direct sum of the Witt groups of shifted local systems on strata. We compare this with another “splitting decomposition” for Witt classes of perverse sheaves obtained inductively from our main new tool, a “splitting relation” which is a generalisation of isotropic reduction.

The Witt groups $W_{\pm }\left(PervX\right)$ are identified with the (nontrivial) Balmer–Witt groups of the constructible derived category $D_c^b\left(X\right)$ of sheaves on $X$, and also with the corresponding cobordism groups defined by Youssin.

Our methods are primarily algebraic and apply more widely. The general context in which we work is that of a triangulated category with duality, equipped with a self-dual $t$-structure with noetherian heart, glued from self-dual $t$-structures on a thick subcategory and its quotient.